MAT2691 - Mathematics II EXAM PREP.

8.8.1 Preparation Paper I (May/June 2010) Question 1 Determine the following integrals: 1.1 2 2 sin x x dx  (2) 1.2 2 2 1 n x dx   (5) 1.3 2 9 4 dx  x   (4) 1.4 2 27 6 x dx  x x    (7) [18] Question 2 2.1 Determine the root-mean-square value (RMS) of 2 1 1 y x   between x  0 and x  3 . (8) 2.2 Calculate the volume of the cone generated by revolving the area between y x  2 , the y -axis and the line y  5 about the y-axis. (6) [14] Question 3 Determine the inverse of the matrix A = 250 37 1 5 31              using the adjoint method. (6) [6] Question 4 4.1 Determine dx dy of the following and simplify if possible. 4.1.1 sin y y xe  (3) 4.1.2 2 1 3 x x x x y   ) (5) 4.1.3   1 y x cosh 2sinh  (4) 4.2 The total area S of a cone with base radius r and perpendicular height h is given by 2 22 S r rr h     . If each of r and h increases at a rate of 2 mm/s,find the rate at which S will increase when r = 40 mm and h = 50 mm. (5) [17] MAT2691/101/3/2020 21 Question 5 5.1 If   x Q x yn y          , show that 5.1.1 Q Q x y Q x y       (3) 5.1.2 2 2 2 2 2 2 Q Q x y x y      (4) 5.2 For the curve given by x  t and   1 2 4 yt t    4 ,0 , find dy dx and 2 2 d y dx at the point ( 2, 3). (7) 5.3 The frequency f associated with any moving particle is given by 2 2 mv f h  Where h is a constant. If the percentage error in the measurement of m is 0.5% and the percentage error in the measurement of v is 0.3%, calculate the maximum percentage error in f. (5) [19] Question 6 6.1 Use the Maclaurin series to expand x e to four non-zero terms. (3) 6.2 Use the answer in question 6.1 to write down a series expansion for x e . (1) 6.3 Use your answers in question 6.1 and 6.2 to write down a series expansion for sinh x to four non-zero terms. (2) [6] TOTAL: 80 22 8.8.2 Preparation Paper II (October 2010) Question 1 Determine the following integrals: 1.1   2 32 ax bx c ax bx cx d dx     236  (2) 1.2 2 2 tan 2 9 cos 2 x dx x      (4) 1.3 sinh 1 cosh x dx  x   (2) 1.4 4 3 3 0 sin .cos x x dx   (5) 1.5 2 4 20 dx x x       (3) [16] Question 2 Determine dx dy of the following and simplify if possible: 2.1   2 2 cos 2 0 x y xy  (4) 2.2 sin . x x y ex  (4) 2.3   1 1 sinh x y x   (4) [12] Question 3 3.1 Find the Maclaurin series expansion of sin x to three terms. (4) 3.2 Use your answer from question 3.1 to write down the Maclaurin expansion for sin . x x (1) 3.3 Determine   0 sin lim . x x x  (1) [6] Question 4 4.1 The area enclosed by the curve 2 y x   4, the x-axis and the lines x 1 and x  4 is rotated about the y- axis. Determine the volume of the solid of revolution produced. (6) 4.2 A beam of length L has a uniform load W given by sin  x W Wo L   where Wo is a constant and x is the distance along the beam. The total load P and the reaction R are given by 0 L P  Wdx  and   0 L R xW dx   .